Journal of Optimization Theory and Applications

, Volume 155, Issue 1, pp 239–251 | Cite as

On Geodesic E-Convex Sets, Geodesic E-Convex Functions and E-Epigraphs

  • Akhlad Iqbal
  • Shahid Ali
  • I. Ahmad


In this paper, we introduce a new class of sets and a new class of functions called geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold. The concept of E-quasiconvex functions on R n is extended to geodesic E-quasiconvex functions on Riemannian manifold and some of its properties are investigated. Afterwards, we generalize the notion of epigraph called E-epigraph and discuss a characterization of geodesic E-convex functions in terms of its E-epigraph. Some properties of geodesic E-convex sets are also studied.


Geodesic E-convex sets Geodesic E-convex functions Geodesic E-quasiconvex function E-epigraphs Riemannian manifolds 



The authors are highly thankful to anonymous referees and the editor for their valuable suggestions/comments which have contributed to the final preparation of the paper.


  1. 1.
    Arana, M., Ruiz, G. Rufian, A. (eds.): Optimality Conditions in Vector Optimization. Bentham Science, Bussum (2010) Google Scholar
  2. 2.
    Boltyanski, V., Martini, H., Soltan, P.S.: Excursions Into Combinatorial Geometry. Springer, Berlin (1997) zbMATHCrossRefGoogle Scholar
  3. 3.
    Danzer, L., Gruenbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Klee, V. (ed.) Convexity. Proc. Sympos. Pure Math., vol. 7, pp. 101–180. Amer. Math. Soc., Providence (1963) Google Scholar
  4. 4.
    Martini, H., Swanepoel, K.J.: Generalized convexity notions and combinatorial geometry. Congr. Numer. 164, 65–93 (2003) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Martini, H., Swanepoel, K.J.: The geometry of minkowski spaces—a survey, Part II. Expo. Math. 22, 14–93 (2004) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hanson, M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Zalinescu, C.: A critical view on invexity. J. Optim. Theory Appl. (2011). (To appear) Google Scholar
  8. 8.
    Yang, X.M., Yang, X.Q., Teo, K.L.: Characterizations and applications of prequasi-invex functions. J. Optim. Theory Appl. 110, 645–668 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Yang, X.M., Yang, X.Q., Teo, K.L.: Generalized invexity and generalized invariant monotonicity. J. Optim. Theory Appl. 117, 607–625 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Antczak, T.: New optimality conditions and duality results of G type in difffierentiable mathematical programming. Nonlinear Anal. 66, 1617–1632 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Noor, M.A., Noor, K.I.: Some characterizations of strongly preinvex functions. J. Math. Anal. Appl. 316, 697–706 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fan, L., Guo, Y.: On strongly α-preinvex functions. J. Math. Anal. Appl. 330, 1412–1425 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Youness, E.A.: On E-convex sets, E-convex functions and E-convex programming. J. Optim. Theory Appl. 102, 439–450 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Duca, D.I., Duca, E., Lupsa, L., Blaga, R.: E-convex functions. Bull. Appl. Comput. Math. 43, 93–103 (2000) Google Scholar
  15. 15.
    Duca, D.I., Lupsa, L.: On the E-epigraph of an E-convex function. J. Optim. Theory Appl. 120, 341–348 (2006) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fulga, C., Preda, V.: Nonlinear programming with E-preinvex and local E-preinvex functions. Eur. J. Oper. Res. 192, 737–743 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Syau, Y.-R., Lee, E.S.: Some properties of E-convex functions. Appl. Math. Lett. 18, 1074–1080 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Yang, X.M.: On E-convex sets, E-convex functions and E-convex programming. J. Optim. Theory Appl. 109, 699–704 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Rapcsak, T.: Smooth Nonlinear Optimization in ℝn. Kluwer Academic, Amsterdam (1997) Google Scholar
  20. 20.
    Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic, Amsterdam (1994) zbMATHGoogle Scholar
  21. 21.
    Pini, R.: Convexity along curves and invexity. Optimization 29, 301–309 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Mititelu, S.: Generalized invexity and vector optimization on differentiable manifolds. Diff. Geom. Dyn. Syst. 3, 21–31 (2001) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Iqbal, A., Ahmad, I., Ali, S.: Strong geodesic α-preinvexity and invariant α-monotonicity on Riemannian manifolds. Numer. Funct. Anal. Optim. 31, 1342–1361 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Ahmad, I., Iqbal, A., Ali, S.: On properties of geodesic η-preinvex functions. Adv. Oper. Res., 10 pp. (2009). Article ID 381831 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  2. 2.Department of MathematicsBITS Pilani Hyderabad CampusHyderabadIndia
  3. 3.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

Personalised recommendations